Harmonic Analysis of Non-Phase-Locked Tides with Red Noise Using the red_tide Package

A novel tidal analysis package (red_tide) has been developed to characterize low-amplitude non-phase-locked tidal energy and dominant tidal peaks in noisy, irregularly sampled, or gap-prone time series. We recover tidal information by expanding conventional harmonic analysis to include prior information and assumptions about the statistics of a process, such as the assumption of a spectrally colored background, treated as nontidal noise. This is implemented using Bayesian maximum posterior estimation and assuming Gaussian prior distributions. We utilize a hierarchy of test cases, including synthetic data and observations, to evaluate this method and its relevance to analysis of data with a tidal component and an energetic nontidal background. Analysis of synthetic test cases shows that the methodology provides robust tidal estimates. When the background energy spectrum is nearly spectrally white, red_tide results replicate results from ordinary least squares (OLS) commonly used in other tidal packages. When background spectra are red (a spectral slope of −2 or steeper), red_tide’s estimates represent a measurable improvement over OLS. The approach highlights the presence of tidal variability and low-amplitude constituents in observations by allowing arbitrarily configurable fitted frequencies and prior statistics that constrain solutions. These techniques have been implemented in MATLAB in order to analyze tidal data with non-phase-locked components and an energetic background that pose challenges to the commonly used OLS approach.